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A pegging algorithm for separable continuous nonlinear knapsack problems with box constraints

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A pegging algorithm for separable continuous nonlinear knapsack problems with box constraints
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  PROOF COVER SHEET Author(s): Chih-Hang WuArticle title: A pegging algorithm for separable continuous nonlinear knapsack problems with box constraintsArticle no: GENO646263Enclosures: 1) Query sheet2) Article proofs Dear Author, 1. Please check these proofs carefully . It is the responsibility of the corresponding author tocheck these and approve or amend them. A second proof is not normally provided. Taylor &Francis cannot be held responsible for uncorrected errors, even if introduced during the production process. Once your corrections have been added to the article, it will be consideredready for publication.Please limit changes at this stage to the correction of errors. You should not make insignificantchanges, improve prose style, add new material, or delete existing material at this stage. Makinga large number of small, non-essential corrections can lead to errors being introduced. 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A full list of the comments andedits you have made can be viewed by clicking on the “Comments” tab in the bottom left-handcorner of the PDF.If you prefer, you can make your corrections using the CATS online correction form.    1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950  Engineering Optimization Vol. 00, No. 0, Month 2012, 1–15 A pegging algorithm for separable continuous nonlinearknapsack problems with box constraints Gitae Kim and Chih-Hang Wu*  Department of Industrial and Manufacturing Systems Engineering, Kansas State University, Manhattan,Kansas, USA (  Received 13 December 2010; final version received 10 November 2011 )This article proposes an efficient pegging algorithm for solving separable continuous nonlinear knapsack problems with box constraints. A well-known pegging algorithm for solving this problem is the Bitran–Hax algorithm, a preferred choice for large-scale problems. However, at each iteration, it must calculatean optimal dual variable and update all free primal variables, which is time consuming. The proposedalgorithm checks the box constraints implicitly using the bounds on the Lagrange multiplier withoutexplicitly calculating primal variables at each iteration as well as updating the dual solution in a moreefficientmanner.Resultsofcomputationalexperimentshaveshownthattheproposedalgorithmconsistentlyoutperforms the Bitran–Hax in all baseline testing and two real-time application models. The proposedalgorithm shows significant potential for many other mathematical models in real-world applications withstraightforward extensions. Keywords: optimization; convex programming; nonlinear programming; nonlinear knapsack problem;pegging algorithm 1. Introduction Theknapsackproblem,alsoknownastheresourceallocationproblem,concernsahitchhikerwhowants to pack his knapsack by selecting from among various possible objects that will give himmaximumcomfort.Thiscanbeformulatedbyamathematicalmodelwheretheobjectivefunctionis to maximize total comfort, with the knapsack constraint being the capacity of the knapsack and binary variables (Martello and Toth 1990). If its objective function is nonlinear, then theproblem is a nonlinear knapsack problem. Different classes of the nonlinear knapsack problemand their reviews have been complied by Bretthauer and Shetty (2002). This article focused on aproblem which has a convex, differentiable and nonlinear objective function, and box constraintsforallvariables.Thisiscalledaconvex,separableandcontinuoustypeofproblem.Therearemanyapplicationsforproblemsofthistype(Robinson etal. 1992)suchasportfolioselection(Markowitz1952), multicommodity network flow (Ali et al. 1980), transportation (Ohuchi and Kaji 1984),support vector machine (Nehate 2006), production planning (Tamir 1980) and convex quadratic *Corresponding author. Email: chw@ksu.edu ISSN 0305-215X print / ISSN 1029-0273 online© 2012 Taylor & Francishttp: // dx.doi.org / 10.1080 / 0305215X.2011.646263http: // www.tandfonline.com Techset Composition Ltd, Salisbury GENO646263.TeX Page#: 15 Printed: 30/12/2011  51525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100 2 G. Kim and C.-H. Wu programming (Dussault et al. 1986). The problem can also be considered as a subproblem formany optimization models. Ibaraki and Katoh (1988) comprehensively discussed the algorithmicaspects of the resource allocation problem and its variants. Twenty years later, Patriksson (2008)surveyed the history and applications of the problem, as well as solving algorithms. These twodiscussions are not reviewed here.Thisarticleconsidersacontinuousnonlinearknapsackproblemwithboxconstraintsasfollows:(P1)  Min n  i = 1  f  i (  x  i ) (1) s . t  . n  i = 1 a i  x  i = b (2) l i ≤ x  i ≤ u i , i = 1, ... , n (3)where f  i (  x  i ) is a nonlinear, convex and differentiable function; the constraint (2) is linear (referredto as the knapsack constraint in the rest of the article); x  i ∈ R , l i ∈ R , u i ∈ R for all i = 1, ... , n , b ∈ R ; and it assumes that all coefficients a i are non-zero, and  ni = 1 a i l i ≤ b ≤  ni = 1 a i u i and  f   i (  x  i ) is invertible.The Lagrangian dual formulation of P1 by relaxing the knapsack constraint (2) is as follows:(D1)  Max  π(α) (4)where π(α) = Min n  i = 1  f  i (  x  i ) + α  n  i = 1 a i  x  i − b  (5) s . t  . l i ≤ x  i ≤ u i , i = 1, ... , n (6)where π(α) is the P1 Lagrange function minimization and α ∈ R is the Lagrange multipliercorresponding to the knapsack constraint (2).The nonlinear knapsack problem P1 is frequently solved iteratively. More than a handful of algorithms have been proposed to solve this problem, and they can be generally divided intotwo main categories (Patriksson 2008): the Lagrange multiplier search method and the variablepegging method. Figures 1 and 2 illustrate these two methods.In Figures 1 and 2, solid boxes denote a variable or variables explicitly used to find the optimalsolution in each algorithm, and dashed boxes represent a variable or variables implicitly opti-mized as the other variable or variables are explicitly optimized.As Bretthauer and Shetty (2002)noted, while the Lagrange multiplier search method maintains all Karush–Kuhn–Tucker (KKT)conditionsduringitsiterations,excepttheknapsackconstraintanditscorrespondingcomplemen-tary slackness condition, the pegging method primal and dual approximations satisfy the KKTconditions in all the iterations, except for the box constraints. That means the multiplier search     C   o    l   o   u   r   o   n    l    i   n   e ,    B    /    W    i   n   p   r    i   n    t One dual variable(Lagrange multiplier) Optimum Primal variables(Implicitly)Search algorithm Figure 1. Lagrange multiplier search method.
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